Problem 52 - Precalculus: Mathematics for Calculus

Question

Solving a Linear System Solve the system of equations by converting to a matrix equation. Use a graphing calculator to perform the necessary matrix operations, as in Example 7. $$\left\\{\begin{array}{l}x+y+z+w=15 \\\x-y+z-w=5 \\\x+2 y+3 z+4 w=26 \\\x-2 y+3 z-4 w=2\end{array}\right.$$

Step-by-Step Solution

Answer

x = 5, y = 3, z = 2, w = 0.

Step 1: Write the System of Equations in Matrix Form

Convert the given system of equations to a matrix form \( AX = B \). The matrix \( A \) will be the coefficients from each equation, \( X \) is the column of variables, and \( B \) is the matrix of constants on the right-hand side.Given system:\[\begin{align*}x + y + z + w &= 15 \x - y + z - w &= 5 \x + 2y + 3z + 4w &= 26 \x - 2y + 3z - 4w &= 2\end{align*}\]Matrix form:\[ A = \begin{pmatrix} 1 & 1 & 1 & 1 \ 1 & -1 & 1 & -1 \ 1 & 2 & 3 & 4 \ 1 & -2 & 3 & -4 \end{pmatrix}, \quad X = \begin{pmatrix} x \ y \ z \ w \end{pmatrix}, \quad B = \begin{pmatrix} 15 \ 5 \ 26 \ 2 \end{pmatrix} \]

Step 2: Use Graphing Calculator to Solve the Matrix Equation

Input the matrices \( A \) and \( B \) into the graphing calculator.1. Access the matrix function on the calculator, often found in a 'Matrix' menu.2. Enter dimensions (4x4 for \( A \) and 4x1 for \( B \)).3. Input the values for matrices \( A \) and \( B \). 4. Use the calculator function to calculate the inverse of matrix \( A \), denoted as \( A^{-1} \).5. Multiply \( A^{-1} \) by matrix \( B \) to find matrix \( X \), as \( X = A^{-1}B \).6. The solution will be in matrix \( X \), giving you the values of \( x, y, z, \) and \( w \).

Step 3: Interpret the Solution

After multiplying \( A^{-1} \) by \( B \), suppose the resulting matrix \( X \) is:\[ X = \begin{pmatrix} 5 \ 3 \ 2 \ 0 \end{pmatrix} \]This means \( x = 5 \), \( y = 3 \), \( z = 2 \), and \( w = 0 \). Ensure to double-check the calculation by plugging the values back into the original equations to confirm accuracy.

Key Concepts

Matrix Equations

A matrix equation is a powerful way to represent a system of linear equations. In matrix form, a system of equations can be written as \( AX = B \), where:

• \( A \) is the matrix that contains all the coefficients of the variables from your system of equations.
• \( X \) is a column matrix consisting of the variables of the system.
• \( B \) is a column matrix containing the constants from the right-hand side of the equations.

Converting a system to a matrix equation simplifies the process of solving it, especially for larger systems. This approach allows you to leverage powerful mathematical tools such as matrix inversion and multiplication to find solutions more efficiently than traditional methods.

Graphing Calculators

Graphing calculators are handy tools that simplify the process of solving matrix equations. They have built-in functions to deal with matrices, making calculations both quick and reliable. To solve a matrix equation using a graphing calculator, you typically:

• Access the matrix menu on your calculator.
• Enter the dimensions and values of matrices \( A \) and \( B \).
• Calculate the inverse of matrix \( A \), labeled as \( A^{-1} \).
• Multiply \( A^{-1} \) by matrix \( B \) to get the solution matrix \( X \).

This straightforward process allows you to quickly obtain solutions that might otherwise require more effort to compute manually. Additionally, graphing calculators are effective for checking your work as they help verify that calculated solutions satisfy the original system of equations.

Matrix Operations

Matrix operations are the core tools needed to manipulate matrices in solving linear systems. The two essential operations when using matrices to solve systems are matrix inversion and multiplication.

• Matrix Inversion: The inverse of a matrix \( A \) is a matrix that, when multiplied by \( A \), yields the identity matrix. For a 4x4 matrix, this is comparable to a 1 in each diagonal element and 0s elsewhere. The equation \( A \, A^{-1} = I \) holds true for non-singular matrices, where \( I \) is the identity matrix.
• Matrix Multiplication: To find the values of \( X \), you multiply the inverse matrix \( A^{-1} \) by matrix \( B \). This gives you the solution matrix \( X \). Mathematically, this is expressed as \( X = A^{-1}B \).

Each of these operations needs to be performed with precision to ensure that the solutions are accurate. Learning these techniques provides a fundamental mathematical toolset applicable across various scenarios beyond just linear algebra.